﻿Electro dynamic Phenomena suggested by Gauss. 453 



integrals are infinitely small ; hence instead of the preceding we 

 may write 



r 



P=22ee'lV(rf Wf'(t, r-f <r). ... (9) 



tio •> 



This expression is distinguished from that given under (8) 

 by the altered form of the function ; and after this change has 

 been effected, Riemann again alters the order of the integrations 

 and thereby obtains 



T 



P= j W22ee' t °Wf'(t, t + ct). . . (10) 



v «/ 



From this point the further calculation is very simple. If the 

 function undei* the second integral sign is developed according 

 to a, we get, neglecting the higher members, 



P= ( ^T22ee'(\/cr[F(T, t) +<tF 1 (t, t)], 



or, by performing the second integration, 



P= f rfrSS^g F'(r, r) + ^F.fr r)] . 



This expression resolves into two members, of which that con- 

 taining F'(t, t) disappears when the summation is performed with 

 respect to e', and there remains 



,.2 



•Jo 



A-SSrfg-p^T.Tjj .... (11) 



* 



or, differently written, 



T =[^^^r dT (12) 



This is the expression deduced by Riemann, which represents 

 the elcctrodynamic action of the two currents upon each other. 



That which I consider to be incorrect in this analysis is the 

 manner in which Riemann exchanges the integrations in order 

 to pass from equation (7) to equation (8), and afterwards from 

 (9) to (10). This exchange would only be admissible in case the 



magnitude - which forms the upper limit of one integral, were 



independent of the time t, according to which the other integra- 

 tion is to take place. This, however, is not the case, but, if the 

 quantities of electricity e and e' move, their distance r is variable 

 with the time. Hence the equations which are obtained by 

 Phil Mag. S. 4. Vol. 37. No. 251. June 1869. 2 H 



